The Random House Dictionary of Popular Proverbs and Sayings states that the expression dates back to the late 1400's. The problem is, it doesn't cite a source for this claim. Some sources hypothesize that the idiom owes something to the widely held belief that the left hand- or foot- was unlucky or less skilled, and so it was encouraged to always lead off with the right (aha! right hand privilege!) Others suggest that this expression is merely literal advice to lead off with whichever foot is stronger when starting on a journey or activity.
It is also found in another, slightly later source: the 1613 poem A Wife by Sir Thomas Overbury which has a line which reads "Hee is still setting the best foot forward."
So that's what is known about the idiom "put your best foot forward." No one is really sure which- if any- of these origin stories is correct, so you can pick which one you like best and go with it.